Solve (1+1/a/a^2+frac{1{a}}:1/a^2/a-1+frac{1{a}})=?

$$( \frac { 1 + \frac { 1 } { a } } { a ^ { 2 } + \frac { 1 } { a } } : \frac { \frac { 1 } { a ^ { 2 } } } { a - 1 + \frac { 1 } { a } } ) = ?$$

Evaluate

$$a$$

Short Solution Steps

Divide \frac{1+\frac{1}{a}}{a^{2}+\frac{1}{a}} by \frac{\frac{1}{a^{2}}}{a-1+\frac{1}{a}} by multiplying \frac{1+\frac{1}{a}}{a^{2}+\frac{1}{a}} by the reciprocal of \frac{\frac{1}{a^{2}}}{a-1+\frac{1}{a}}.

$$\frac{\left(1+\frac{1}{a}\right)\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$\frac{\left(\frac{a}{a}+\frac{1}{a}\right)\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a+1}{a}\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a}{a}.

$$\frac{\frac{a+1}{a}\left(\frac{\left(a-1\right)a}{a}+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

Since \frac{\left(a-1\right)a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a+1}{a}\times \frac{\left(a-1\right)a+1}{a}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

Do the multiplications in \left(a-1\right)a+1.

$$\frac{\frac{a+1}{a}\times \frac{a^{2}-a+1}{a}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

Multiply \frac{a+1}{a} times \frac{a^{2}-a+1}{a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2} times \frac{a}{a}.

$$\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\left(\frac{a^{2}a}{a}+\frac{1}{a}\right)\times \frac{1}{a^{2}}}$$

Since \frac{a^{2}a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{2}a+1}{a}\times \frac{1}{a^{2}}}$$

Do the multiplications in a^{2}a+1.

$$\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{3}+1}{a}\times \frac{1}{a^{2}}}$$

Multiply \frac{a^{3}+1}{a} times \frac{1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{3}+1}{aa^{2}}}$$

Divide \frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa} by \frac{a^{3}+1}{aa^{2}} by multiplying \frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa} by the reciprocal of \frac{a^{3}+1}{aa^{2}}.

$$\frac{\left(a+1\right)\left(a^{2}-a+1\right)aa^{2}}{aa\left(a^{3}+1\right)}$$

Cancel out aa in both numerator and denominator.

$$\frac{a\left(a+1\right)\left(a^{2}-a+1\right)}{a^{3}+1}$$

Factor the expressions that are not already factored.

$$\frac{a\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a+1\right)\left(a^{2}-a+1\right)}$$

Cancel out \left(a+1\right)\left(a^{2}-a+1\right) in both numerator and denominator.

$$a$$

Differentiate w.r.t. a

$$1$$

Steps Using Definition of a Derivative

Divide \frac{1+\frac{1}{a}}{a^{2}+\frac{1}{a}} by \frac{\frac{1}{a^{2}}}{a-1+\frac{1}{a}} by multiplying \frac{1+\frac{1}{a}}{a^{2}+\frac{1}{a}} by the reciprocal of \frac{\frac{1}{a^{2}}}{a-1+\frac{1}{a}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(1+\frac{1}{a}\right)\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(\frac{a}{a}+\frac{1}{a}\right)\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a+1}{a}\left(a-1+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a+1}{a}\left(\frac{\left(a-1\right)a}{a}+\frac{1}{a}\right)}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

Since \frac{\left(a-1\right)a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a+1}{a}\times \frac{\left(a-1\right)a+1}{a}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

Do the multiplications in \left(a-1\right)a+1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a+1}{a}\times \frac{a^{2}-a+1}{a}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

Multiply \frac{a+1}{a} times \frac{a^{2}-a+1}{a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\left(a^{2}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2} times \frac{a}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\left(\frac{a^{2}a}{a}+\frac{1}{a}\right)\times \frac{1}{a^{2}}})$$

Since \frac{a^{2}a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{2}a+1}{a}\times \frac{1}{a^{2}}})$$

Do the multiplications in a^{2}a+1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{3}+1}{a}\times \frac{1}{a^{2}}})$$

Multiply \frac{a^{3}+1}{a} times \frac{1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa}}{\frac{a^{3}+1}{aa^{2}}})$$

Divide \frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa} by \frac{a^{3}+1}{aa^{2}} by multiplying \frac{\left(a+1\right)\left(a^{2}-a+1\right)}{aa} by the reciprocal of \frac{a^{3}+1}{aa^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(a+1\right)\left(a^{2}-a+1\right)aa^{2}}{aa\left(a^{3}+1\right)})$$

Cancel out aa in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a\left(a+1\right)\left(a^{2}-a+1\right)}{a^{3}+1})$$

Factor the expressions that are not already factored in \frac{a\left(a+1\right)\left(a^{2}-a+1\right)}{a^{3}+1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a+1\right)\left(a^{2}-a+1\right)})$$

Cancel out \left(a+1\right)\left(a^{2}-a+1\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a)$$

The derivative of ax^{n} is nax^{n-1}.

$$a^{1-1}$$

Subtract 1 from 1.

$$a^{0}$$

For any term t except 0, t^{0}=1.

$$1$$